Polynomial integrability of natural two-dimensional Hamiltonian systems and applications

我们的项目研究具有两个自由度的可积哈密顿系统。我们研究了动量多项式积分的存在性。我们打算应用新开发的方法来构建这种系统的新的物理有趣的例子，并解决一些著名的猜想。特别是，我们计划解决归因于Birkhoff和Whittaker问题的两个猜想。我们的目标是研究这类系统对磁流的自然推广，并将所得结果应用于Finsler几何和广义相对论。进一步，我们研究了磁台球和Finsler台球的可积性。

Our project deals with integrable Hamiltonian systems with two degrees of freedom. We study the existence of integrals which are polynomial in momenta. We intend to apply new developed methods in order to construct new phyiscally interesting examples of such systems and tackle a number of famous conjectures. In particular, we plan to solve two conjectures attributed to Birkhoff and the problem of Whittaker. We aim to study the natural generalisations of such systems for magnetic flows and apply obtained results in Finsler geometry and general relativity. Further, we study integrability of magnetic and Finsler billiards.